nLab amnestic isofibration

Idea

A functor is an amnestic isofibration if it is an amnestic functor and an isofibration. The combination of these properties admits a particularly simply description in terms of a lifting property.

Proposition

A functor $U : \mathcal{D} \to \mathcal{C}$ is an amnestic isofibration if and only if, for any object $D$ in $\mathcal{D}$ and any isomorphism $f : C \to U D$ in $\mathcal{C}$, there is a unique isomorphism $\tilde{f} : \tilde{C} \to D$ such that $U \tilde{f} = f$.

Proof

Indeed, if $\tilde{f}' : \tilde{C}' \to D$ were any other isomorphism such that $U \tilde{f}' = f$, then $U (\tilde{f}^{-1} \circ \tilde{f}') = id_C$, so we must have $\tilde{f} = \tilde{f}'$.

Amnestic isofibrations are occasionally called discrete isofibrations, but this term may be misleading, because they are not isofibrations with discrete fibres.

Properties

Proposition

A monadic functor is strictly monadic if and only if it is also an amnestic isofibration.

Proof

Clearly, a strictly monadic functor is an amnestic isofibration; and if a monadic functor $U$ is amnestic, then the comparison functor $K$ is also amnestic, and if $U$ is a monadic isofibration, so is $K$; therefore in this case $K$ must be an isomorphism of categories.

Examples

• If $F : A \to B$ is a bijective on objects functor, then $[F, C] : [B, C] \to [A, C]$ is an amnestic isofibration (Lemma 2.4.8 of Avery 2017).

Related concepts

• amnestic functor
• isofibration
• conservative functor
• monadic functor

References

A reference using the term amnestic isofibrations:

• Tom Avery: Structure and Semantics [arXiv:1708.01050]

A reference using the term discrete isofibrations:

• Steve Lack: A 2-categories companion, in John Baez, Peter May: Towards Higher Categories, Springer, (2009) [arXiv:math/0702535]